Remainder and Factor Theorem

The polynomial division algorithm is as follows: suppose $d(x)$ and $p(x)$ are nonzero polynomials where the degree of $p(x)$ is greater than or equal to the degree of $d(x)$. Then there exist two unique polynomials, $q(x)$ and $r(x)$, such that $p(x)=d(x)q(x)+r(x)$, where either $r(x)=0$ or the degree of $r(x)$ is strictly less than the degree of $d(x)$.

Remainder Theorem: Suppose $p(x)$ is a polynomial of degree at least 1 and c is a real number. When $p(x)$ is divided by $(x-c)$ the remainder is $p(c)$.

Proof: By the division algorithm, $p(x)=(x-c)q(x)+r$, where r must be a constant since $d(x)=x-c$ has a degree of 1. $p(x)=(x-c)q(x)+r$ must hold for all values of $x$, so we can set $x=c$ and get that $p(c)=(c-c)q(x)+r=r$. Thus the remainder $r=p(c)$.

Factor Theorem: Suppose $p(x)$ is a nonzero polynomial. The real number $c$ is a zero of $p(x)$ if and only if $(x-c)$ is a factor of $p(x)$.

By the division algorithm, $x-c$ is a factor of $p(x)$ if and only if $r=0$. So, since $p(c)=r$ when $p(x)$ is divided by $x-c$, $x-c$ is a factor of $p(x)$ if and only if $p(c)=0$; that is, if $c$ is a zero of $p(x)$.

Resources

• The Factor Theorem and Remainder Theorem, Mathematics LibreTexts
• Dividing Polynomials, Paul's Online Notes